Gradient bounds for strongly singular or degenerate parabolic systems

Abstract

We consider weak solutions $u:\Omega_{T}\rightarrow\mathbb{R}^{N}$ to parabolic systems of the type \[ u_{t}-\mathrm{div}\,A(x,t,Du)=f \qquad \mathrm{in}\ \Omega_{T}=\Omega\times(0,T), \] where $\Omega$ is a bounded open subset of $\mathbb{R}^{n}$ for $n\geq2$, $T>0$ and the datum $f$ belongs to a suitable Orlicz space. The main novelty here is that the partial map $\xi\mapsto A(x,t,\xi)$ satisfies standard $p$-growth and ellipticity conditions for $p>1$ only outside the unit ball $\{\vert\xi\vert<1\}$. For $p>\frac{2n}{n+2}$ we establish that any weak solution \[ u\in C^{0}((0,T);L^{2}(\Omega,\mathbb{R}^{N}))\cap L^{p}(0,T;W^{1,p}(\Omega,\mathbb{R}^{N})) \] admits a locally bounded spatial gradient $Du$. Moreover, assuming that $u$ is essentially bounded, we recover the same result in the case $1<p\leq\frac{2n}{n+2}$ and $f=0$. Finally, we also prove the uniqueness of weak solutions to a Cauchy-Dirichlet problem associated with the parabolic system above. We emphasize that our results include both the degenerate case $p\geq2$ and the singular case $1<p<2$.